Quantum Corrections in Nanoplasmonics: Shape, Scale, and Material

Abstract

The classical treatment of plasmonics is insufficient at the nanometer-scale due to quantum mechanical surface phenomena. Here, an extension of the classical paradigm is reported which rigorously remedies this deficiency through the incorporation of first-principles surface response functions-the Feibelman d parameters-in general geometries. Several analytical results for the leading-order plasmonic quantum corrections are obtained in a first-principles setting; particularly, a clear separation of the roles of shape, scale, and material is established. The utility of the formalism is illustrated by the derivation of a modified sum rule for complementary structures, a rigorous reformulation of Kreibig's phenomenological damping prescription, and an account of the small-scale resonance shifting of simple and noble metal nanostructures. DOI: 10.1103/PhysRevLett.118.157402 Classical treatments of plasmonics require specification of just two elements: geometry, involving shape and scale, and dielectric environment, supplied through local bulk dielectric functions. In the deep subwavelength regime, i.e., in the nonretarded limit, even the element of scale is rendered superfluous by scale-invariant governing equations. As the geometric scale is reduced further, below 10-20 nm in metals, toward the intrinsic quantum mechanical length scales of the plasmon-supporting electron gas, the classical approach inevitably deteriorates, as established by numerous experiments [1][2][3][4][5][6][7][8][9][10]. The main shortcomings of the classical approach can be divided into three categories [11], resulting from the neglect of (i) spill-out of the conduction electron's wave function beyond the material boundaries [12], (ii) nonlocality, i.e., the momentum dependence of the bulk response functions [13], and (iii) incomplete accounting of internal electron dynamics, especially surface-enabled plasmon damping by electron-hole pair creation [10,14]. In the subnanometer domain, additional shortcomings are expected to materialize, e.g., due to size quantization [15,16] and the breakdown of jellium treatments [17,18]. Jointly, these shortcomings and their impact on plasmonic observables (modal spectrum, field enhancements, local density of states, etc.) motivate and define the field of quantum nanoplasmonics [12,13,17].While time-dependent density-functional theory (TDDFT) [19], in principle, can bridge the gap between classical and quantum nanoplasmonics, its explicit application is, in practice, limited to few-atom clusters and systems of high spatial symmetry due to computational constraints. A sizable fraction of nanoplasmonic structures of interest [20][21][22], thus, fall in a region which is simultaneously inaccessible to conventional, explicit TDDFT and beyond the validity of classical plasmonics, roughly spanning characteristic geometric scales L ∼ 2-20 nm. In this Letter, we provide a simple and general answer to the central question raised by this dichotomy: namely, what are the leading-order nonclassical corrections to cla...

Landau damping of surface plasmons in metal nanostructures

“…2 and 3. Such effects are best described within the TDLDA approach [52][53][54][55][56][57][58][59] and are out of scope of this paper.…”

“…This dependence was, in fact, masked in all previous quantum-mechanical studies of simple-shape systems [44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59], where a specific functional form of the local field, appropriate for the given geometry, was adopted, while it is completely missing in the CS approach. Moreover, for the most widely studied spherical geometry, the local field is uniform inside the particle (apart from surface effects), which further obscured its importance.…”

“…Surface-assisted plasmon decay (Landau damping) has been extensively studied experimentally [33][34][35][36][37][38][39][40][41][42][43] and theoretically [44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59] since the pioneering paper by Kawabata and Kubo [44], who have shown that, for a spherical particle of radius a, the surface scattering rate is γ sp = 3v F /4a, where v F is the electron Fermi velocity. In subsequent quantum-mechanical studies carried within random phase approximation (RPA) [45][46][47][48][49][50][51] and timedependent local density approximation (TDLDA) [52][53][54][55][56][57][58][59] approaches, a more complicated picture has emerged involving the role of confining potential and nonlocal effects.…”

“…In subsequent quantum-mechanical studies carried within random phase approximation (RPA) [45][46][47][48][49][50][51] and timedependent local density approximation (TDLDA) [52][53][54][55][56][57][58][59] approaches, a more complicated picture has emerged involving the role of confining potential and nonlocal effects. These are dominant at the spatial scale ξ nl = v F /ω that defines the characteristic length for nonlocal effects [60,61] (e.g., for noble metals, v F /ω < 1 nm in the plasmon frequency range), whereas for larger systems with L ≫ v F /ω (i.e., several nm and larger), they mainly affect the overall magnitude of γ sp , while preserving intact its size dependence [56,58].…”

“…The major roadblock in the way of carrying this program forward has so far been the lack of any quantummechanical model for evaluation of γ s in a nanostructure of arbitrary shape. Due to the complexity of electronic states in general-shape confined systems, calculations of γ s were performed, within RPA [44][45][46][47][48][49][50][51] and TDLDA [52][53][54][55][56][57][58][59] approaches, only for some simple (mostly spherical) geometries. For general shape systems, the surface scattering rate was suggested, within the classical scattering (CS) model [62][63][64][65][66], in the form γ cs = Av F /L, where L is interpreted as the ballistic scattering length in a classical cavity, while the phenomenological constant A accounts for the effects of surface potential, electron spillover, and dielectric environment.…”